Optimal. Leaf size=20 \[ \left (2+x+3 e^{-3+\frac {e^5 x}{3}} x\right )^2 \]
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Rubi [B] time = 0.22, antiderivative size = 101, normalized size of antiderivative = 5.05, number of steps used = 21, number of rules used = 5, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 1593, 2196, 2176, 2194} \begin {gather*} 6 e^{\frac {e^5 x}{3}-3} x^2+9 e^{\frac {2 e^5 x}{3}-6} x^2-36 e^{\frac {e^5 x}{3}-8} x+12 e^{\frac {e^5 x}{3}-3} x-36 e^{\frac {e^5 x}{3}-8}+(x+2)^2+36 e^{\frac {e^5 x}{3}-8} (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1593
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (e^6 (4+2 x)+e^{\frac {2 e^5 x}{3}} \left (18 x+6 e^5 x^2\right )+e^{\frac {e^5 x}{3}} \left (e^3 (12+12 x)+e^8 \left (4 x+2 x^2\right )\right )\right ) \, dx}{e^6}\\ &=(2+x)^2+\frac {\int e^{\frac {2 e^5 x}{3}} \left (18 x+6 e^5 x^2\right ) \, dx}{e^6}+\frac {\int e^{\frac {e^5 x}{3}} \left (e^3 (12+12 x)+e^8 \left (4 x+2 x^2\right )\right ) \, dx}{e^6}\\ &=(2+x)^2+\frac {\int e^{\frac {2 e^5 x}{3}} x \left (18+6 e^5 x\right ) \, dx}{e^6}+\frac {\int \left (12 e^{3+\frac {e^5 x}{3}} (1+x)+2 e^{8+\frac {e^5 x}{3}} x (2+x)\right ) \, dx}{e^6}\\ &=(2+x)^2+\frac {\int \left (18 e^{\frac {2 e^5 x}{3}} x+6 e^{5+\frac {2 e^5 x}{3}} x^2\right ) \, dx}{e^6}+\frac {2 \int e^{8+\frac {e^5 x}{3}} x (2+x) \, dx}{e^6}+\frac {12 \int e^{3+\frac {e^5 x}{3}} (1+x) \, dx}{e^6}\\ &=36 e^{-8+\frac {e^5 x}{3}} (1+x)+(2+x)^2-\frac {36 \int e^{3+\frac {e^5 x}{3}} \, dx}{e^{11}}+\frac {2 \int \left (2 e^{8+\frac {e^5 x}{3}} x+e^{8+\frac {e^5 x}{3}} x^2\right ) \, dx}{e^6}+\frac {6 \int e^{5+\frac {2 e^5 x}{3}} x^2 \, dx}{e^6}+\frac {18 \int e^{\frac {2 e^5 x}{3}} x \, dx}{e^6}\\ &=-108 e^{-13+\frac {e^5 x}{3}}+27 e^{-11+\frac {2 e^5 x}{3}} x+9 e^{-6+\frac {2 e^5 x}{3}} x^2+36 e^{-8+\frac {e^5 x}{3}} (1+x)+(2+x)^2-\frac {18 \int e^{5+\frac {2 e^5 x}{3}} x \, dx}{e^{11}}-\frac {27 \int e^{\frac {2 e^5 x}{3}} \, dx}{e^{11}}+\frac {2 \int e^{8+\frac {e^5 x}{3}} x^2 \, dx}{e^6}+\frac {4 \int e^{8+\frac {e^5 x}{3}} x \, dx}{e^6}\\ &=-108 e^{-13+\frac {e^5 x}{3}}-\frac {81}{2} e^{-16+\frac {2 e^5 x}{3}}+12 e^{-3+\frac {e^5 x}{3}} x+6 e^{-3+\frac {e^5 x}{3}} x^2+9 e^{-6+\frac {2 e^5 x}{3}} x^2+36 e^{-8+\frac {e^5 x}{3}} (1+x)+(2+x)^2+\frac {27 \int e^{5+\frac {2 e^5 x}{3}} \, dx}{e^{16}}-\frac {12 \int e^{8+\frac {e^5 x}{3}} \, dx}{e^{11}}-\frac {12 \int e^{8+\frac {e^5 x}{3}} x \, dx}{e^{11}}\\ &=-108 e^{-13+\frac {e^5 x}{3}}-36 e^{-8+\frac {e^5 x}{3}}-36 e^{-8+\frac {e^5 x}{3}} x+12 e^{-3+\frac {e^5 x}{3}} x+6 e^{-3+\frac {e^5 x}{3}} x^2+9 e^{-6+\frac {2 e^5 x}{3}} x^2+36 e^{-8+\frac {e^5 x}{3}} (1+x)+(2+x)^2+\frac {36 \int e^{8+\frac {e^5 x}{3}} \, dx}{e^{16}}\\ &=-36 e^{-8+\frac {e^5 x}{3}}-36 e^{-8+\frac {e^5 x}{3}} x+12 e^{-3+\frac {e^5 x}{3}} x+6 e^{-3+\frac {e^5 x}{3}} x^2+9 e^{-6+\frac {2 e^5 x}{3}} x^2+36 e^{-8+\frac {e^5 x}{3}} (1+x)+(2+x)^2\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.11, size = 58, normalized size = 2.90 \begin {gather*} 2 \left (2 x+\frac {x^2}{2}+\frac {9}{2} e^{-6+\frac {2 e^5 x}{3}} x^2+e^{\frac {e^5 x}{3}} \left (\frac {6 x}{e^3}+\frac {3 x^2}{e^3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 42, normalized size = 2.10 \begin {gather*} {\left (9 \, x^{2} e^{\left (\frac {2}{3} \, x e^{5}\right )} + {\left (x^{2} + 4 \, x\right )} e^{6} + 6 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {1}{3} \, x e^{5} + 3\right )}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 107, normalized size = 5.35 \begin {gather*} \frac {1}{2} \, {\left (2 \, {\left (x^{2} + 4 \, x\right )} e^{6} + 9 \, {\left (2 \, x^{2} e^{10} - 6 \, x e^{5} + 9\right )} e^{\left (\frac {2}{3} \, x e^{5} - 10\right )} + 27 \, {\left (2 \, x e^{5} - 3\right )} e^{\left (\frac {2}{3} \, x e^{5} - 10\right )} + 12 \, {\left (x^{2} e^{10} + 2 \, x e^{10} - 6 \, x e^{5} - 6 \, e^{5} + 18\right )} e^{\left (\frac {1}{3} \, x e^{5} - 7\right )} + 72 \, {\left (x e^{5} + e^{5} - 3\right )} e^{\left (\frac {1}{3} \, x e^{5} - 7\right )}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 43, normalized size = 2.15
method | result | size |
risch | \(x^{2}+4 x +9 x^{2} {\mathrm e}^{-6+\frac {2 x \,{\mathrm e}^{5}}{3}}+\left (6 x^{2} {\mathrm e}^{3}+12 x \,{\mathrm e}^{3}\right ) {\mathrm e}^{-6+\frac {x \,{\mathrm e}^{5}}{3}}\) | \(43\) |
norman | \(\left (x^{2} {\mathrm e}^{3}+4 x \,{\mathrm e}^{3}+12 x \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}+6 x^{2} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}+9 x^{2} {\mathrm e}^{-3} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}}\right ) {\mathrm e}^{-3}\) | \(55\) |
derivativedivides | \(3 \,{\mathrm e}^{-6} {\mathrm e}^{-5} \left ({\mathrm e}^{6} \left (\frac {x^{2} {\mathrm e}^{5}}{3}+\frac {4 x \,{\mathrm e}^{5}}{3}\right )+54 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}} x}{6}-\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}}}{4}\right )+54 \,{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}} x^{2}}{18}-\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}} x}{6}+\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}}}{4}\right )+12 \,{\mathrm e}^{3} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}+12 \,{\mathrm e}^{3} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x \,{\mathrm e}^{5}}{3}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}\right )+36 \,{\mathrm e}^{3} {\mathrm e}^{-5} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x \,{\mathrm e}^{5}}{3}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}\right )+18 \,{\mathrm e}^{3} {\mathrm e}^{-5} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x^{2} {\mathrm e}^{10}}{9}-\frac {2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x \,{\mathrm e}^{5}}{3}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}\right )\right )\) | \(212\) |
default | \({\mathrm e}^{-6} \left ({\mathrm e}^{6} \left (x^{2}+4 x \right )+162 \,{\mathrm e}^{-5} \left ({\mathrm e}^{-5} \left (\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}} x}{6}-\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}}}{4}\right )+{\mathrm e}^{-5} \left (\frac {{\mathrm e}^{10} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}} x^{2}}{18}-\frac {{\mathrm e}^{5} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}} x}{6}+\frac {{\mathrm e}^{\frac {2 x \,{\mathrm e}^{5}}{3}}}{4}\right )\right )+3 \,{\mathrm e}^{-5} \left (12 \,{\mathrm e}^{3} {\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}+12 \,{\mathrm e}^{3} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x \,{\mathrm e}^{5}}{3}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}\right )+36 \,{\mathrm e}^{3} {\mathrm e}^{-5} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x \,{\mathrm e}^{5}}{3}-{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}\right )+18 \,{\mathrm e}^{3} {\mathrm e}^{-5} \left (\frac {{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x^{2} {\mathrm e}^{10}}{9}-\frac {2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}} x \,{\mathrm e}^{5}}{3}+2 \,{\mathrm e}^{\frac {x \,{\mathrm e}^{5}}{3}}\right )\right )\right )\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 45, normalized size = 2.25 \begin {gather*} {\left (9 \, x^{2} e^{\left (\frac {2}{3} \, x e^{5}\right )} + {\left (x^{2} + 4 \, x\right )} e^{6} + 6 \, {\left (x^{2} e^{3} + 2 \, x e^{3}\right )} e^{\left (\frac {1}{3} \, x e^{5}\right )}\right )} e^{\left (-6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.66, size = 33, normalized size = 1.65 \begin {gather*} x\,{\mathrm {e}}^{-6}\,\left ({\mathrm {e}}^3+3\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{3}}\right )\,\left (4\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3+3\,x\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.19, size = 51, normalized size = 2.55 \begin {gather*} x^{2} + 4 x + \frac {9 x^{2} e^{3} e^{\frac {2 x e^{5}}{3}} + \left (6 x^{2} e^{6} + 12 x e^{6}\right ) e^{\frac {x e^{5}}{3}}}{e^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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